Problem: Simplify the following expression and state the condition under which the simplification is valid. $r = \dfrac{x^2 - 49}{x + 7}$
Solution: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = x$ $ b = \sqrt{49} = 7$ So we can rewrite the expression as: $r = \dfrac{({x} + {7})({x} {-7})} {x + 7} $ We can divide the numerator and denominator by $(x + 7)$ on condition that $x \neq -7$ Therefore $r = x - 7; x \neq -7$